【作者】 黄小为；

【导师】 吴传生；

【作者基本信息】 武汉理工大学 ， 应用数学， 2004， 硕士

【摘要】 反问题是现在数学物理研究中的一个热点问题，但是数学物理反问题的求解面临的一个本质性的困难是不适定性，主要是近似解的不稳定性，即方程的解(如果存在)不连续依赖于右端的数据，当右端的数据有误差时，其解与真解之间会产生很大的误差。求解不适定问题的普遍方法是正则化方法，如何建立有效的正则化方法及算法是反问题领域中不适定问题研究的重要内容。 本文从一些实例出发，介绍了反问题和不适定问题的基本概念，并讨论了方程的Moore-Penrose广义解和Moore-Penrose广义逆，得出了线性紧算子方程的不适定性，即Moore-Penrose广义解的不稳定性的结论。为了得到线性紧算子方程稳定的近似解，介绍了不适定问题正则化的一般理论，以自伴紧算子的谱分析与紧算子奇异值分解为理论基础，利用奇异系给出了解的表达式，说明了紧算子方程不适定性的根源在于紧算子的奇异值趋于零的性质，由此通过引入正则化滤子函数来减弱或滤掉奇异值趋于零的性质对解的稳定性的影响，构造正则算子，从而提供了建立正则化方法的理论依据。文中以此为依据给出了一个正则化滤子函数，从而建立一种新的正则化方法，并讨论了正则解的误差估计及正则参数的选取问题，证明了这种方法使正则解的误差具有渐进最优阶。文中还介绍了两种重要的正则化滤子函数，并讨论了其对应的Tikhonov正则化和Landweber迭代法，这两种方法避免了奇异系的计算，在各类反方问题的研究中被广泛地采用。考虑到反问题的数值计算需要将问题离散化，化为有限维的问题来进行处理，而对于有限维算子的奇异系的计算已经有了相当成熟的各种算法，因此在反问题的数值计算中没有必要避开奇异系的计算，此时TSVD(谱截断)正则化方法是十分简单并相当有效的正则化方法。文中详细讨论了TSVD正则解的误差估计与正则参数的选取问题，通过正则参数的先验和后验选取，证明了TSVD正则解的误差具有渐进最优阶，并且通过具体的实例说明了TSVD正则化方法是一种计算量小，正则参数容易确定的求解不适定问题十分有效的方法。更多还原

【Abstract】 In mathematical physics inverse problems’ researching is very hot nowadays. However, the main difficulty about the solution of inverse problems lies in ill-posedness, which is the instability of approximate solution, that is, the solution of a equation (if existing) incontinuously rely on the data in right hand side. There will produce a great error between the approximate solution and the correct solution where the data in right hand side are errant. A general way that we solute ill-posed problems is regularization method. Therefore, how to build up effective regularization method and algorithm are very important parts of ill-pose problems researching in inverse problems field.Beginning with some cases, the article gives basic definitions of inverse problems and ill-posed problems. Then it discusses the Moore-Penrose generalized solution and the Moore-Penrose generalized inverse, and makes a conclusion that linear compact operator equations are ill-posed, that is, the Moore-Penrose generalized solution is unstable. In order to find a stable approximate solution of linear compact operator equation, the article introduces general theories about ill-posed problems, it bases on spectral theory of self-adjiont compact operators and the singular value decomposition for compact operators, avails singular system to give expression of the solution, and explains ill-posedness of compact operator equation roots in the property that the singular values trends to zero. Thereout, it is provided with theoretic support of building up regularization method by inducting regularization filter to weaken or filtrate the influence that the nature of the singular value being very close to zero has on the solution’s stability. On the basis of that, the article gives a regularization filter and builds up a quite new method of regularization. It also discusses the calculation of regularization resolution’s enor and the choice of regularization parameter, and proves that this method is the best to make the resolution have order optimality. Besides, the article presents two important regularization filter and also discusses corresponding Tikhonov Regularization and Landweber Iterative Method. These two methods avoid calculating singular system and are well applied in the fields of all kinds of inverse problems’ researching. Note that regularized problems are usually defined in an infinite setting and have to be discTetized for an implementation and there have been many comparatively adult methods of calculating singular system. Therefore, there is no need to avoid the calculation of singular system in inverse problems’ calculating of numeric value. Here TSVD Regularization Method is very simple and quite effective one. The article discusses the calculation of TSVD regularization resolution’s error and the choice of TSVD regularization parameter in details. By prior and posterior choices of regularization parameter, it is proved that the error of regularization solution has order optimality, and through cases it is also explained that TSVD Regularization Method is a effective one to resolve ill-pose problems with characteristics of little amount of calculating and of easier regularization parameter confirming.更多还原